The route to chaos

Let us return to Fig. 39, which tracks the evolution of a left-favouring
periodic attractor as the quality-factor is gradually increased. Recall that when
exceeds a critical value, which is about , then the attractor
undergoes a period-doubling bifurcation which converts it from a period-1 to a period-2
attractor. This bifurcation is indicated by the forking of the curve in
Fig. 39. Let us now investigate what happens as we continue to increase . Fig. 43 is basically
a continuation of Fig. 39. It can be seen that, as is gradually
increased, the attractor undergoes a period-doubling bifurcation at , as before,
but then undergoes a second period-doubling bifurcation (indicated by the
second forking of
the curves) at , and a third bifurcation at . Obviously, the second bifurcation
converts a period-2 attractor into a period-4 attractor (hence, two curves split apart to
give four curves). Likewise, the third bifurcation converts a period-4 attractor
into a period-8 attractor (hence, four curves split into eight curves).
Shortly after the third
bifurcation, the various curves in the figure seem to expand explosively and merge together
to produce an area of almost solid black. As we shall see, this behaviour is indicative
of the onset of chaos.

Figure 44 is a blow-up of Fig. 43, showing more details of the onset of
chaos. The period-4 to period-8 bifurcation can be seen quite clearly. However,
we can also see a period-8 to period-16 bifurcation, at
. Finally,
if we look carefully, we can see a hint of a period-16 to period-32 bifurcation, just
before the start of the solid black region.
Figures 43 and 44 seem to suggest that the
onset of chaos is triggered by an infinite series of period-doubling bifurcations.

Table 2 gives some details of the sequence of period-doubling bifurcations
shown in Figs. 43 and 44. Let us introduce a bifurcation index : the
period-1 to period-2 bifurcation corresponds to ; the
period-2 to period-4 bifurcation corresponds to ; and so on. Let be
the critical value of the quality-factor above which the th bifurcation is
triggered. Table 2 shows the , determined from Figs. 43 and 44, for
to 5. Also shown is the ratio

(98)

for to 5. It can be seen that Tab. 2 offers reasonably
convincing evidence that this ratio takes the constant value .
It follows that we can estimate the critical -value required to trigger the th
bifurcation via the following formula:

(99)

for . Note that the distance (in ) between bifurcations decreases rapidly as
increases. In fact, the above
formula predicts an accumulation of period-doubling bifurcations at , where

(100)

Note that our calculated accumulation point corresponds almost exactly to the onset of the
solid black region in Fig. 44.
By the time that exceeds , we expect the attractor to have been
converted into a period-infinity
attractor via an infinite series of period-doubling bifurcations.
A period-infinity attractor is one whose corresponding motion never repeats
itself, no matter
how long we wait. In dynamics, such bounded aperiodic motion is generally referred to as chaos.
Hence, a period-infinity attractor is sometimes called a chaotic attractor.
Now, period- motion is represented by separate curves in Fig. 44.
It is, therefore, not surprising that chaos (i.e., period-infinity motion) is
represented by an infinite number of curves which merge together to form a region of solid black.

Table 2:The period-doubling cascade.

Bifurcation

period-1period-2

-

-

period-2period-4

0.02133

-

period-4period-8

0.00455

period-8period-16

0.00097

period-16period-32

0.00020

Let us examine the onset of chaos in a little more detail. Figures 45-48
show details of the pendulum's time-asymptotic motion at various stages on the
period-doubling cascade discussed above. Figure 45 shows period-4 motion:
note that the Poincaré section consists of four points, and the associated sequence of
net rotations per period of the pendulum repeats itself every four periods.
Figure 46 shows period-8 motion:
now the Poincaré section consists of eight points, and the rotation sequence
repeats itself every eight periods. Figure 47 shows period-16 motion:
as expected, the Poincaré section consists of sixteen points, and the rotation sequence
repeats itself every sixteen periods. Finally, Fig. 48 shows chaotic motion. Note that
the Poincaré section now consists of a set of four continuous line segments, which are, presumably, made
up of an infinite number of points (corresponding to the infinite period of chaotic motion).
Note, also, that the associated sequence of net rotations per period shows no obvious sign of ever
repeating itself. In fact, this sequence looks rather like one of the previously shown periodic
sequences with the addition of a small random component. The generation of apparently
random motion from equations of motion, such as Eqs. (81) and
(82), which contain no overtly random elements is one of the most surprising features of
non-linear dynamics.

Many non-linear dynamical systems, found in nature, exhibit a transition
from periodic to chaotic motion as some control parameter is varied.
Now, there are various known mechanisms by which chaotic motion can arise from periodic
motion. However, a transition to chaos via an infinite series of period-doubling bifurcations, as
illustrated
above, is certainly amongst the most commonly occurring of these mechanisms.
Around 1975, the physicist Mitchell Feigenbaum was investigating a simple mathematical
model, known as the logistic map, which exhibits a transition to chaos,
via a sequence of period-doubling bifurcations, as a control parameter
is increased. Let be the value of at which the
first -period cycle appears. Feigenbaum noticed that the ratio

(101)

converges rapidly to a constant value, , as increases. Feigenbaum was
able to demonstrate that this value of is common to a wide range of different
mathematic models which exhibit transitions to chaos via period-doubling
bifurcations.28Feigenbaum went on to argue that the Feigenbaum ratio, ,
should converge to the value in any dynamical system exhibiting a
transition to chaos via period-doubling bifurcations.29 This amazing prediction has
been verified experimentally in a number of quite different physical
systems.30 Note that our best estimate of the
Feigenbaum ratio (see Tab. 2) is , in good agreement with Feigenbaum's
prediction.

The existence of a universal ratio characterizing the transition to chaos via
period-doubling bifurcations is one of many pieces of evidence indicating that
chaos is a universal phenomenon (i.e., the onset and nature
of chaotic motion in different dynamical systems has many common features).
This observation encourages us to believe that in studying the chaotic motion of
a damped, periodically driven, pendulum we are learning lessons which can
be applied to a wide range of non-linear dynamical systems.